Design of Shewhart-type control charts with estimated parameter for the Rayleigh distribution using frequentist and Bayesian approaches

Studies on control charts with estimated parameters have received much attention in the recent literature. In this paper, the effect of parameter estimation on the performance of the Shewhart-type chart for the Rayleigh distribution, namely the V-R chart, is first studied under the conditional perspective. It is found that parameter estimation has a serious effect on the performance of the frequentist V-R chart. In order to solve this problem, the frequentist V-R chart is adjusted by using the exceedance probability criterion to guarantee the in-control performance. Since the frequentist VR chart uses the sample information from Phase I, but not the process information from past experience, an alternative chart, namely the Bayesian V-R chart, is proposed based on the predictive distribution of the plotting statistic. The performances of the Bayesian and adjusted frequentist V-R charts are evaluated and compared in terms of the percentiles, mean, and standard deviation of the conditional average run length distribution. The results suggest that the Bayesian chart outperforms the frequentist counterpart, especially when more prior information is available.